[Rick TYler]

Originally Posted by ChrisH

Then break up any rectangular open spaces into triangles by attaching something diagonally across the corners.

A lightweight sheer plate (like the 5-6mm luaun plywood sold at some home centers) is also an amazingly effective way to stiffen a structure. My teams must have built a dozen competition robots in the last five years with two vertical towers tied together with a lightweight stiffening plate between them. It's like magic, although Chris' x-bracing also works well if you can find the right connectors.

(Most of the advice in this thread is really good. I'm just tossing in another approach.) The other advantage of a stiffening panel is that it gives you a nice flat surface to display sponsors and your robot's name.